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Assume . give 0 = ortho- (1.1) ing V δ 2 MICHAELEHRIGANDCATHARINASTROPPEL was shown that the Brauer algebra can be equipped with a nonnegative Z- grading (which is even Koszul in case δ 6= 0), see [ES1], [ES2], [Li]. As a consequence of our theorem, the endomorphism rings (1.1) inherit a grading. The existence of such a grading on Brauer algebras is rather unexpected and up to now this grading cannot be described intrinsically in terms of the representation theory of the Lie superalgebra. The existence of a ring homomorphism (1.1) from the Brauer algebra to the endomorphism ring of V ⊗d is not new. It is for instance a special case of the Brauer algebra actions defined in [BCHLLS] and also follows from the more abstract theory of Deligne categories, [De]. However, the isomorphism theorem is, as far as we know, new. The crucial point here is that in general (and in particular in contrast to the classical case), the tensor space is not completely reducible and then unfortunately the methods from [GW] and [BCHLLS] do not apply. As far as we know there is not much known about the indecomposable summands appearing in this tensor space and even less is known about the general structure of the category of finite dimensional representations of osp(V ), see [GS], [Sea], and also [Co] for some partial results. This is very much in contrast to the case of the general Lie superalgebra gl(m|n), where the corresponding (mixed) tensor spaces were recently studied and described in detail, [CW], [BS2]. There, the role of the Brauer algebra is played by the walled Brauer algebra respectively walled Brauer category, sometimes also called oriented Brauer category, see [BS2], [BCNR], [SS]. As our main tool we construct an embedding of the Brauer algebra into an additive closure of the walled Brauer category and then use the above mentioned results for the general linear case to deduce Theorem A in the ortho-symplectic case. More precisely we restrict the action of osp(V ) to a suitably embedded gl(m|n) and decompose the occurring representation V ⊗d as a direct sum of mixed tensor products on which the walled Brauer category acts naturally. One of the crucial steps is the following generalization from [BS2]:
Theorem B. Let m,n ∈ Z≥0, d > 0. The corresponding oriented Brauer ⊗d category OBd(m − n) acts on V and its action commutes with the action of gl(m|n). We then embed the Brauer algebra into the endomorphism ring of V ⊗d viewed as an object inside the additive closure of the oriented Brauer cate- gory. The injectivity of Theorem A is then deduced from the corresponding injectivity result [BS2, Theorem 7.1] for walled Brauer algebras. Finally we show (by very elementary arguments) that any gl(m|n) endomorphism that commutes also with the action of osp(V ) is already contained in the image of the embedding and gives the desired isomorphism. (This method of proof does not give optimal bounds. Optimal bounds could be established recently in [LZ3], however on the cost of using less elementary arguments.) We also like to stress that our proof does not rely on a Schur-Weyl du- ality for the ortho-symplectic supergroup as considered independently by SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 3
Lehrer and Zhang in [LZ1] using methods from graded-commutative alge- braic geometry. Our proof of the isomorphism theorem is considerably more elementary and only relies on a rather simple reduction to the general linear Lie superalgebra case, see (5.13) and the mixed Schur-Weyl duality from [BS2] which in turn is deduced from the Schur-Weyl duality of Sergeev [Sev] and Berele-Regev [BR] for the general linear Lie superalgebras. A different reduction argument (to the general linear group) was recently used in [DLZ]. Using however the non-elementary grading results of [BS2], it is possible to describe also the kernel of our action in general, but it requires to work with a graded version of the Brauer algebra as defined in [ES1] and will appear in a separate article where this graded Brauer algebra is studied. The extra grading refines the results from [LZ1], [LZ2]. We expect that our approach generalizes easily to the quantized (super) case using the quantised walled Brauer algebras
Acknowledgement We thank Gus Lehrer for pointing out an inaccuracy in an earlier version of the paper as well as Kevin Coulembier and Daniel Tubbenhauer for various comments on a preliminary version.
2. The Lie superalgebra We fix as ground field the complex numbers C. By a superspace we mean a Z/2Z -graded vector space V = V0 ⊕ V1; for any homogeneous element v ∈ V we denote by |v| ∈ {0, 1} its parity. The integer dim V0 − dim V1 is called the supertrace of V and we denote by sdimV = dim V0| dim V1 its superdimension. Given a superspace V let gl(V ) be the corresponding general Lie superalgebra, i.e. the superspace EndC(V ) of all endomorphism with the superbracket [X,Y ]= X ◦ Y − (−1)|X|·|Y |Y ◦ X.
For the whole paper we fix now n ∈ Z≥0 and a finite dimensional super- space V = V0 ⊕ V1 with dim V1 = 2n, and equipped with a non-degenerate supersymmetric bilinear form h−, −i, i.e. a bilinear form V × V → C which is symmetric when restricted to V0 × V0, skew-symmetric on V1 × V1 and zero on mixed products. It will sometimes be convenient to work with a fixed homogeneous basis vi, i ∈ I of V , for a suitable indexing set I, and ∗ ∗ right dual basis vi , i.e. vi, vj = δi,j. Attached to this data we have the following: D E Definition 2.3. The ortho-symplectic Lie superalgebra osp(V ) is the Lie supersubalgebra of gl(V ) consisting of all endomorphisms which respect the supersymmetric bilinear form. Explicitly, a homogeneous element X ∈ osp(V ) has to satisfy hXv, wi + (−1)|X|·|v| hv, Xwi = 0, (2.2) for homogeneous v ∈ V . 4 MICHAELEHRIGANDCATHARINASTROPPEL
From now on we make the convention that, whenever the parity of an element appears in a formula, the element is assumed to be homogeneous. Remark 2.4. If X ∈ osp(V ), then hXv, wi= 6 0 implies |X| = |v| + |w|.
3. The Brauer algebra The following algebra was originally defined by Brauer [Br] in his study of the orthogonal group.
Definition 3.5. Let d ∈ Z≥0 and δ ∈ C. The Brauer algebra Brd(δ) is the associative unital C-algebra generated by the elements
si, ei for 1 ≤ i ≤ d − 1, subject to the relations 2 si = 1, sisj = sjsi, sksk+1sk = sk+1sksk+1, 2 ei = δei, eiej = ejei, ekek+1ek = ek+1, ek+1ekek+1 = ek, (3.3) siei = ei = eisi, skek+1ek = sk+1ek, sk+1ekek+1 = skek+1, for 1 ≤ i, j ≤ d − 1, | i − j |> 1, and 1 ≤ k ≤ d − 2.
Following Brauer we realize Brd(δ) as a diagram algebra: A Brauer di- agram on 2d vertices is a partitioning b of the set {1, 2,...,d, 1∗, 2∗,...d∗} into d subsets of cardinality 2. Let B[d] be the set of such Brauer diagrams. An element can be displayed graphically by arranging 2d vertices in two rows 1, 2,...,d and 1∗, 2∗,...,d∗, with each vertex linked to precisely one other vertex. Two such diagrams are considered to be the same or equivalent if they link the same d pairs of points. Special Brauer diagrams are the ”unit” 1 = {{1, 1∗}, {2, 2∗}, · · · , {d, d∗}} connecting always j with j∗ for all 1 ≤ j ≤ r, and for 1 ≤ i ≤ r − 1 ∗ the si (respectively ei) which connects j with j except of the new pairs {i, (i + 1)∗}, {i + 1, i∗} (respectively {i, i + 1}, {i∗, (i + 1)∗}) involving the numbers i and i + 1;
i i+1 i i+1
si ei (3.4) Given two Brauer diagrams b and b′, their concatenation b′ ◦ b is obtained by putting b on top of b′ identifying vertex i∗ in b with vertex i in b′ and removing all the internal loops. Let c(b, b′) be the number of loops removed. Then we have the following fact, see e.g. [GW, Section 9 and 10] for details:
Lemma 3.6. The Brauer algebra Brd(δ) is, via the assignment (3.4) on generators, canonically isomorphic to the C-algebra with basis B[d] and mul- ′ tiplication bb′ = δc(b,b )b ◦ b′ for b, b′ ∈B[d].
Remark 3.7. Generically, the algebra Brd(δ) is semi-simple, but not semi- simple for specific integral values for δ (dependent on d), see [We], [Ru] and SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 5 also [AST]. For a detailed study of the non semi-simple algebras over the complex numbers we refer to [ES1, 2.2]. We now define the two actions on the tensor space V ⊗d from Theorem A. The action of osp(V ) on V ⊗d is given by the comultiplication ∆(X)= X ⊗ 1 + 1 ⊗ X for X ∈ osp(V ), keeping in mind the tensor product rule (X ⊗ Y )(v ⊗ w) = (−1)|Y |·|v|(Xv ⊗ Y w).
Explicitly we have for homogeneous elements wi ∈ V r i−1 (P |wj |)|X| X.(w1⊗ . . . ⊗wr)= (−1) j=1 w1⊗ . . . ⊗wi−1⊗Xwi⊗wi+1⊗ . . . ⊗wr. i X=1 Definition 3.8. Define the following linear endomorphisms σ : V ⊗ V −→ V ⊗ V , via σ(v ⊗ w) := (−1)|v|·|w|w ⊗ v and
|vi| ∗ τ : V ⊗ V −→ V ⊗ V , via τ(v ⊗ w) := hv, wi (−1) vi ⊗ vi . i I X∈ of V ⊗ V and for each fixed natural number d ≥ 2 the endomorphisms ⊗(i−1) ⊗(d−i−1) ⊗(i−1) ⊗(d−i−1) si = id ⊗ σ ⊗ id and ei = id ⊗ τ ⊗ id of V ⊗d for 1 ≤ i < d. The following proposition is straight-forward to check.
Proposition 3.9. The maps σ, τ and si, ei are osp(V )-equivariant.
Theorem 3.10. Let δ = dimV0 − dimV1 be the supertrace of V . Then there ⊗d is a right action of Brd(δ) on V given by
v.si := si(v) and v.ei = ei(v) for v ∈ V ⊗d and 1 ≤ i ≤ d − 1. It commutes with the action of osp(V ).
Remark 3.11. Restricting only to the si gives us an action of the symmetric ⊗d group Sd on V , which clearly commutes with the action of gl(V ). This is the super Schur-Weyl duality, [BS2, Theorem 7.5], originally proved by Sergeev [Sev] and Berele-Regev, [BR]. Proof of Theorem 3.10. By Proposition 3.9 all involved morphisms commute with the action of osp(V ). Hence it remains to show that it defines an action of the Brauer algebra. It is obvious that the si define an action of the symmetric group and that the relations eiej = ejei are satisfied for 2 |i − j| > 1. Moreover, ei = δei by definition. To verify the relation siei = ei it suffices to assume d = 2. Let v, w ∈ V be homogeneous then
|v|·|w| |v|·|w| |vl| ∗ (v ⊗ w).siei = (−1) (w ⊗ v).ei = (−1) hw, vi (−1) vl ⊗ vl l I X∈ |vl| ∗ = hv, wi (−1) vl ⊗ vl = (v ⊗ w).ei. l I X∈ 6 MICHAELEHRIGANDCATHARINASTROPPEL
∗ For the equality ei = eisi we note that (vj ⊗ vk).ei = 0 unless j = k, so we assume equality and obtain
∗ |vl| ∗ |vl| |vl| ∗ (vj ⊗ vj ).eisi = (−1) (vl ⊗ vl ).si = (−1) (−1) vl ⊗ vl Xl∈I Xl∈I |vl| |vl| ∗ |vl| ∗ = (−1) ((−1) vl ) ⊗ vl = (−1) vl ⊗ vl , Xl∈I Xl∈I |vl| ∗ where the final equality holds because the (−1) vl also form a basis of V with right dual basis {vl}. For the final relations
eiei+1ei(x)= ei(x) and ei+1eiei+1(x)= ei+1(x), (3.5)
eiei+1si(x)= eisi+1(x) and ei+1eisi+1(x)= ei+1si(x) (3.6) for x ∈ V ⊗d, it suffices to consider the case d = 3 and x = u ⊗ v ⊗ w for homogeneous u, v, w ∈ V . The exact calculation is straight-forward and left to the reader.
4. The oriented Brauer Category We now recall the oriented Brauer category (or sometimes called walled Brauer category, see e.g. [SS]), which has as objects certain orientation sequences and morphism spaces given by oriented generalized Brauer dia- grams. We then relate it to the space V ⊗d. We set OSeq[d] = {∧, ∨}×d and OSeq[[ d] = {∧, ∨, ◦}×d and call its ele- ments orientations of length d respectively generalized orientations of length d. Let B[d] be the set of generalized Brauer diagrams on 2d vertices, that means diagrams which are Brauer diagrams, except that the partitioning is into subsetsb of cardinality 1 or 2. In other words, we allow vertices that are not connected to any other vertex. Definition 4.12. An oriented generalized Brauer diagram is a triple (t, b, s) where s, t ∈ OSeq[[ d] and b ∈ B[d] such that the following conditions hold:
(1) if {i, j}∈ b with i 6= j then {ti, tj} = {∧, ∨}, ∗ ∗ (2) if {i , j }∈ b with i 6=bj then {si, sj} = {∧, ∨}, ∗ (3) if {i, j }∈ b then {ti, sj } ∈ {{∧}, {∨}}, and ∗ (4) if {i}∈ b or {i }∈ b, then ti = ◦, respectively si = ◦. The pair (s, t) is then called the orientation. t We denote the set of diagrams oriented with orientation (s, t) by Bs or by t B[d]s if we want to stress that the length is d, and draw them as oriented di- agrams (in the obvious sense) with the orientation sequence s at theb bottom andb the orientation sequence t on the top. [ [ ′ It will be sometimes helpful to vary d and so we set OSeq = d′≥0 OSeq[d ] [ ′ s [ s [ and OSeq = d′≥0 OSeq[d ]. For ∈ OSeq we denote by ∈SOSeq ⊂ OSeq the associated reduced sequence obtained by deleting all ◦’s. Similarly define S SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 7
t t b for b ∈ B[d]. Note that if b ∈ Bs then b ∈ Bs. Moreover, we can allow t pairs of sequences (s, t) not of the same length and define the sets Bs and t Bs of (reduced)b oriented generalizedb Brauer diagramsb with orientation (s, t) exactly as in Definition 4.12. b b (∨,∧,∨,◦) Example 4.13. The first diagram below is an element in B[4](∨,◦,∧,∨). The other two diagrams are not oriented. In both cases the pieces highlighted by a dashed line violate the orientation conditions. b
If one removes the highlighted parts of the diagrams above one obtains ele- (◦) (∨,∧,∨,◦) ments in B(◦,∧,∨) (for the second diagram) and B(∨,∧,∨) (for the third dia- gram). b b Definition 4.14. Let d ∈ Z≥0 and δ ∈ C. We define the oriented Brauer category OBd(m − n) as the following C-linear category: The set of objects is OSeq[[ d], the morphism space Hom(s, t) for s, t ∈ OSeq[[ d] is the vector t space with basis B[d]s, and the composition of morphisms Hom(s, t) × Hom(r, s) −→ Hom(r, t), b is done on basis vectors by putting the two diagrams on top of each other, glueing along the entries of s and eliminating all internal ◦’s and also inner circles, each elimination of an internal circle resulting in multiplying the diagram with a factor of m − n. Similarly, define the category OB(m − n) by allowing as objects sequences of arbitrary finite length. [ Lemma 4.15. Let s, t ∈ OSeq[d] and ℓ = |{i | si = ◦}| − |{i | ti = ◦}|. Then Hom(s, t)= {0} if ℓ is odd or ℓ is even and additionally it holds ℓ 6= |{i | s = ∧}| − |{i | t = ∧}|. 2 i i Proof. This follows directly from the definitions. Remark 4.16. We like to stress that our category OB(m − n) is slightly different from the category called oriented walled Brauer category in [BCNR] in the sense that we have more objects, since we also fix the places of the trivial tensor factors. Hence, the oriented Brauer category from [BCNR] is the full subcategory of ours, where we only allow objects without ◦’s. To relate this category to the endomorphisms of the superspace V ⊗d let m be such that dim(V0) = 2m or dim(V0) = 2m + 1. Unfortunately we have to distinguish between these two cases. Therefore, we will refer to them as the even case respectively odd case. Fix the sets I∧ = {i | 1 ≤ i ≤ m + n}, I∨ = {i | 1 ≤ i ≤ m + n}, I◦ = {0}, 8 MICHAELEHRIGANDCATHARINASTROPPEL and also I = I∧ ∪ I∨ in the even case and I = I∧ ∪ I∨ ∪ I◦ in the odd case. Denote by || i || the absolute value for an element in i ∈ I, in particular || i ||= i =|| i || for i ∈ I∨. To simplify calculations later on we use an explicit realization of osp(V ) in terms of matrices, see e.g. [Mu].
2m+1|2n Definition 4.17. Let V = C be the superspace with basis vi for i ∈ I and |vi| = 0 if || i ||≤ m and |vi| = 1 otherwise. To write down matrices we order the basis elements as follows
0 < 1 < ... < m< 1 < ... A, a1, a2 are m × m matrices T T D, d1, d2 are n × n matrices 0 −u1 −u2 x1 x2 u , u are m × 1 matrices u −AT a CT z 1 2 2 1 1 x1,x2 are 1 × n matrices u1 a2 A z2 B . T T T T B, z1, z2 are m × n matrices −x2 −B −z1 −D d1 C is a n × m matrix T T x1 z2 C d2 D a1, a2 are skew-symmetric d , d are symmetric 1 2 We denote by osp(2m|2n) the Lie superalgebra produced in the same way but 2m|2n using the superspace C obtained by omitting the basis vector v0 instead of V , and then using the matrices obtained by deleting the first row and column in the description of osp(2m + 1|2n). Definition 4.17 is in the odd case consistent with the previous definition of osp(V ) if we take the bilinear form h−, −i given by the following block matrix where ½k denotes the identity matrix of size k × k: 1 0 0 0 0 0 0 ½m 0 0 J = 0 ½m 0 0 0 , 0 0 0 0 −½n 0 0 0 ½n 0 In the even case one has to delete the first row and column. To relate the action of osp(2m + 1|2n) on V ⊗d to the oriented Brauer category we will restrict the action to the general linear Lie supersubalgebra gl(m|n). The embedding ι : gl(m|n) ֒→ osp(2m + 1|2n) is obtained by only allowing non-zero entries in the matrices A, B, C, and D in the notations above; similarly in the even case. These are all embeddings as Lie super subalgebras obtained by omitting one vertex in the Dynkin diagram (high- lighted by a dotted frame). With respect to the distinguished root system for the Cartan subalgebra consisting of diagonal matrices, see [Zh, A.2.1], SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 9 this is given as follows: osp(2m + 1|2n) # # ⊗ # # # # (m>0) m−1 n−1 osp(1|2n) # # # n−1 # osp(2m|2n) # # ⊗ # # (m>1) m−1 n−1 # osp(2|2n) ⊗ # # # # n−1 In the (omitted) classical cases of the orthogonal and symplectic Lie algebra the embedding is the obvious one. Then the restriction of V to the Lie super subalgebra is given by the followings lemma. Lemma 4.18. As a gl(m|n) module V = W∧ ⊕ W∨ ⊕ W◦ in the odd case and V = W∧ ⊕ W∨ in the even case, with ∧ ∨ W∧ = span{vi | i ∈ I }, W∨ = span{vi | i ∈ I }, and W◦ = span{v0}. Moreover, W∧ is isomorphic to the natural representation of gl(m|n), W∨ to its dual and W◦ to the trivial representation. Proof. Clearly, W∧ is isomorphic to the natural representation of gl(m|n). The dual of the natural representation can be described in an appropriate (|i|+|j|)|i| basis wi by Ei,jwk = −δi,k(−1) wj, and so it is isomorphic to W∨ via wi 7→ vi. The statement for W◦ is evident. More generally we obtain the following as a direct consequence: Corollary 4.19. The space V ⊗d decomposes as a gl(m|n)-module as ⊗d ∼ V = Ws1 ⊗ . . . ⊗ Wsd . (4.7) \ s∈OSeq[M d] [ We abbreviate Ws = Ws1 ⊗ . . . ⊗ Wsd and define for each s ∈ OSeq[d] in the odd case (respectively each s ∈ OSeq[d] in the even case) the set of s-indices sj Vect(s)= {i = (i1,...,id) | ij ∈ I }. Obviously this set labels the standard basis vectors in the summand corre- sponding to s in the decomposition (4.7) with vi = vi1 ⊗ . . . ⊗ vid . Lemma 4.20. Let s, t ∈ OSeq[[ d] or s, t ∈ OSeq[d] be two sequences, such that there exists σ ∈ Sd with si = tσ(i). Then σ induces an isomorphism ψσ : Ws −→ Wt, (4.8) of gl(m|n)-modules via the action of Sd given in Remark 3.11. Choosing σ to be of minimal length defines a distinguished isomorphism ψσ. 10 MICHAELEHRIGANDCATHARINASTROPPEL Proof. The action of the symmetric group on V ⊗d even commutes with gl(V ) and hence gives a morphism, and then clearly an isomorphism, of gl(m|n)- modules when restricted to the subspaces. As a consequence we have canonical isomorphisms of algebras ∼ Endgl(m|n)(Ws) = Endgl(m|n)(Wt) for each pair s, t as above, namely given by conjugation with ψσ, where σ is the unique choice of minimal length. Note that these endomorphism rings were described in detail in [BS2]. In case V has large enough dimension (in the sense that d < (m + 1)(n + 1)) and t = (∧,..., ∧, ∨,..., ∨, ◦,..., ◦), where the symbols ∧ and ∨ appear exactly r respectively s times, this en- domorphism ring is exactly the walled Brauer algebra from [BS2, Theorem 7.8] originally introduced [Tu], [KM] and studied in [Ni]. [ Lemma 4.21. Let s, t ∈ OSeq[d] and ℓ = |{i | si = ◦}|−|{i | ti = ◦}|. Then Homgl(m|n)(Ws,Wt)= {0} if ℓ is odd or ℓ is even and then additionally ℓ 6= |{i | s = ∧}| − |{i | t = ∧}|. (4.9) 2 i i Proof. Let ht(λ) denote the height of a gl(m|n) weight, i.e. the sum of all coefficients written with respect to the standard {εi}-basis. In this basis the weights occurring in W∧ are ε1,...,εm+n, those in W∨ are −ε1,..., −εm+n, and in W◦ only the weight 0 occurs. If ℓ is odd, then the height of the weights occurring in Ws will either all be even or all be odd, while in Wt it will be the other way around, thus there can’t be a non-trivial gl(m|n)-morphism. If ℓ is even and (4.9) holds define s∧ = |{i | si = ∧}| and analogously s∨, t∧, and t∨. Assume now s∧ − t∧ > ℓ/2, then it follows that s∨ − t∨ < ℓ/2. By adjointness and Lemma 4.20 we know that s t t s ∼ ⊗( ∧+ ∨) ⊗( ∧+ ∨) Homgl(m|n)(Ws,Wt) = Homgl(m|n)(W∧ ,W∧ ). Then s∧ + t∨ < t∧ + t∨ − ℓ/2 < t∧ + s∨, and so the height of any weight appearing in the domain is strictly less than the height of those appearing in the codomain and so all gl(m|n) morphism are trivial. The remaining case s∧ − t∧ < ℓ/2 is done in exactly the same way. To extend the action from Theorem 3.10 to the oriented Brauer category we need the notion of a weight of an oriented Brauer diagram: [ t Definition 4.22. Let s, t ∈ OSeq. Assume b ∈ B[d]s and i ∈ Vect(s) and j j ∈ Vect(t). We denote by bi the diagram b with the vertices at the bottom labelled by the elements i1,...,id and the verticesb on the top by j1,...,jd j (read always from left to right). We say that bi is consistently labelled if ∗ the labelling function || L(−) || defined by L(k) = jk and L(k ) = ik, for 1 ≤ k ≤ d, is constant on the subsets of the partition b. SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 11 Example 4.23. The first of the following diagrams is consistently labelled, the second and third are not (the highlighted parts are violating the condi- tions): 1 1 2 0 1 1 3 0 1 1 2 0 0 3 0 3 2 3 2 3 2 3 2 3 (4.10) j j Definition 4.24. The weight wt bi of a labelled oriented diagram bi is defined as follows: j j |c| c(−1) h(−1) h′ (−1), if bi is consistently labelled, wt bi := , ( 0, otherwise, Q Q Q where, viewed as diagrams, (1) the first product is over all crossings c of two different strands with the notation |c| = |i| · |j| if the two strands are labelled by i and j; (2) the second product is over all labelled clockwise caps h with large labels, i.e. oriented horizontal strands at the bottom of the diagram with left endpoint oriented ∧ and with labelling set {a, a} such that || a ||>m; (3) the third product is over all labelled anticlockwise cups h′ with large labels, i.e. oriented horizontal strands at the top of the diagram with left endpoint oriented ∨ and with labelling set {a, a} such that || a ||>m. Remark 4.25. Note that the weight of a diagram depends on V . For instance the first diagram in (4.10) has weight 0 if m ≥ 3, has weight −1 if m = 1, 2 and has weight 1 if m = 0. Note that the weight of a crossing is by definition independent of the four possible orientations: t1 = t2 = t3 = t4 = (4.11) On the other hand, we can “rotate” each crossing by applying a cup and a cap on top of the diagram and compute then its weight. But in any case the weights of the added cup and cap multiply to 1, and so the weights don’t change. In fact, we will see in the proof of Theorem 4.27 that the weight is an invariant on equivalence classes of oriented labelled diagrams. By a representation of OBd(m − n) we mean a linear functor F from OBd(m − n) to the category of complex vector spaces. Equivalently we say s OBd(m − n) acts on X := OSeq[\ d] F ( ). If X is moreover an R-module for some ring R, then the action of OB (m − n) commutes with the action L d of R if F (f)(r.x) = r.F (f)(x) for any x ∈ X, r ∈ R and morphism f in OBd(m − n). 12 MICHAELEHRIGANDCATHARINASTROPPEL [ t Definition 4.26. For s, t ∈ OSeq and b ∈ Bs we define the linear map j F (b) : Ws −→ Wt, via F (b)(vib) := wt bi vj. t j∈Vect(X ) on basis vectors vi, where i ∈ Vect(s). Theorem 4.27. Let δ be the supertrace of V . The assignment s 7→ F (s) := Ws and f 7→ F (f), on objects respectively morphisms defines a functor from OB(m − n) to the category of finite dimensional gl(m|n)-modules. It restricts to an action of the oriented Brauer category OBd(m − n) on V ⊗d which commutes with the action of gl(m|n). The identity of the object [ s ∈ OSeq[d] acts by projecting onto the summand Ws. Proof. By definition, the identity of the object s acts by projecting onto the summand Ws. To see that the action is well-defined note that two t diagrams D, D ∈ Bs are the same if and only if their reductions D, D′ are the same. Moreover, F (D) = F (D′) if and only if F (D) = F (D′). ′ t ′ u ′ u ′ Finally assume D1 ∈b Bs, D2 ∈ Bt , and D ∈ Bs then F (D2 · D1)= F (D ) ′ if F (D2 · D1) = F (D ) Hence, to see that the action is well-defined and compatible with multiplicationb itb is enough to restrictb to reduced sequences. By Remark 4.16 it suffices then to show that the oriented Brauer category in the sense of [BCNR] acts. Luckily in this case we have explicit (monoidal) generators and relations for the morphisms, namely the generators c = d = s = (4.12) with all possible labellings and the relations (1.4)-(1.9) in [BCNR]. We only need to keep track of the weights: The first two of these relations amount to the fact that a consistently oriented and labelled kink built from a cup and cap has weight 1, the third just requires that (−1)|i||j|(−1)|i||j| = 1 for any labels i, j, the fourth just requires (−1)|i||j|(−1)|i||k|(−1)|j||k| = (−1)|j||k|(−1)|i||k|(−1)|i||j| and the relation [BCNR, (1.8)] amounts to the fact that the weight of a crossing does not depend on the orientation, but only on the labels, see Remark 4.25. Finally for [BCNR, (1.9)] it is enough to see that the sum of the weights of the diagram d · t3 · c over all labellings equals the supertrace δ. But the labellings contributing weight 1 are precisely those with absolute value at most m, hence there are precisely m. The other n possible labelling always create a cup of weight one and a cap of weight −1 or vice versa. Hence, the total weight is δ = m − n. Hence we obtain an action when restricted to reduced sequences and therefore the claim follows. SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 13 5. The Isomorphism Theorem In this section we prove Theorem A from the introduction. The main step is to establish a commuting diagram of the form ⊗d Endosp(m′|2n) V / Endgl(m|n) ( s Ws) (5.13) O✤ 3 O ❣❣❣❣❣ Ψ ✤ ❣❣❣❣ ΘL ❣❣❣❣❣ ✤ ❣❣❣❣❣ ❣❣❣❣ Φ Brd(δ) / EndMat(OBd(m−n)) ( s s) where δ is the supertrace of V , m′ = 2m +1 or m′ = 2Lm and Φ a map constructed from the action of OB(m − n) from Theorem 4.27. For this we view the algebra Brd(δ) as a category with one object, called ⋆, and endomorphism ring Brd(δ), and construct a functor into Mat(OB(m − n)), the additive closure of OB(m − n). Definition 5.28. The category Mat(OB(m − n)) is defined as follows: The objects of Mat(OB(m−n)) are formal finite direct sums of objects in OB(m− n), and the morphisms are matrices of morphisms between the summands with addition and composition given by the usual rules of matrix multiplica- tion, see e.g. [B-N] for more details. The idea is now to construct a functor which sends the one object of Brd(δ) to the direct sum of all s ∈ OSeq[[ d] and a Brauer diagram b to a matrix with rows and columns indexed by OSeq[[ d]. It will be convenient to write a matrix A as s,t 1tA1s. Diagrammatically we will write the morphisms as formal sums of oriented diagrams, where the orientation sequences stand for the matrix idempotents,P see e.g. (5.14). Due to the fact that we also have the ”place holder” symbols ◦ in the sequences we need the following additional notion: Definition 5.29. Let b, b′ ∈ B[d] be generalized Brauer diagrams.Then b′ ∈ B[d] is called a subdiagram of b ∈ B[d], denoted b′ b, if b′ refines the partition of b. In other words,b the diagram b′ is obtained from b by removing someb arcs with their orientations and replace them with the appropriate number of ◦’s. This allows us to define the functor from Brd(δ) to Mat(OB(m − n)). Proposition 5.30. The assignment Ψ : ⋆ 7→ s and b 7→ M(b), \ s∈OSeq[M d] with M(b)(t,s) ∈ HomOB(m−n)(s, t) equal to the unique subdiagram of b in t B[d]s or zero if such a diagram does not exist, defines a faithful functor from Brd(δ) to Mat(OB(m − n)). b 14 MICHAELEHRIGANDCATHARINASTROPPEL Proof. Clearly, the required subdiagram of b is unique in case it exists. That the functor is faithful is obvious by definition. To see that the functor is well-defined it remains to verify the relations (3.3). The element si is sent to the sum of all oriented crossings at place i, i + 1 plus its subdiagrams obtained by removing one or two strands in the crossing, for instance for d = 2: M(si)= + + + + + + + . (5.14) If d > 2, then the assignment looks locally as above with the remaining strands oriented in all possible ways. Note that at vertex i and i + 1 (resp. i∗ and (i + 1)∗) every possible combination from {∧, ∨, ◦} occurs precisely once and composing Ψ(si) with itself gives the identity. The other two braid relations can also be checked easily. The image of ei looks locally as follows: M(ei)= + + + + + + + + If we compose it with itself then we obtain the same sum, but each oriented diagram with an additional clockwise circle, an additional anticlockwise cir- cle and in the odd case also the original diagram itself. Hence, we obtain δ times the original diagram as required. The fifth relation in (3.3) is clear. For the next relation note that Ψ(ei)Ψ(ei+1)Ψ(ei) is a sum of oriented dia- grams of the same underlying shape as the one for eiei+1ei or subdiagrams of this form, but equipped with orientations. Note that each fixed pair of orientation at the bottom and the top appears in precisely one summand and so Ψ preserves the relation eiei+1ei = ei and similarly also ei+1eiei+1 = ei+1. One can easily check that it also preserves eisi = ei = siei and the last two relations of (3.3). Thus, we have now two actions of the Brauer algebra on V ⊗d, one by Theorem 3.10 and another one given by Proposition 5.30 and Theorem 4.27. Lemma 5.31. The actions of the Brauer algebra given in Theorem 3.10 and the one given by Theorem 4.27 via the embedding of Proposition 5.30 agree. Proof. This follows directly from the definitions by a direct calculation on the generators. By Proposition 5.30 we have the induced map Φ from the Brauer alge- bra to the algebra EndMat(OBd(m−n)) ( s s). Moreover we have the action map Θ from End ( s) to End V ⊗d and know from Mat(OBd(m−n)) s L gl(m|n) Lemma 5.31 that the image of Θ ◦ Φ is contained in End V ⊗d . L osp (2m+1|2n) Hence we get the induced map Ψ as indicated in (5.13). We claim that Ψ is an isomorphism if d ≤ m + n. First note that it is injective, since Φ is SCHUR-WEYL DUALITY FOR THE BRAUER ALGEBRA 15 injective by definition, and Θ is injective if d < (m + 1)(n + 1) by [BS2, Theorem 7.8], in particular it is injective if d ≤ m + n. Our strategy to prove surjectivity will be to use the surjectivity of Θ from [BS2, Theorem 7.8] and show that any element of EndMat(OBd(m−n)) ( s s) that commutes with the action of the ortho-symplectic Lie superalgebra is already contained in the image of Φ. L We will show this by an inductive argument. For this we subdivide our set of generalized Brauer diagrams into smaller sets: B[d]= B[d](k,r), 1≤k≤ad,1≤r≤d b b where B[d](k,r) denotes the set of diagrams with exactly 2r singleton subsets and k vertical strands, i.e. subsets of the form {i, j∗} for some i, j. Fur- thermore,b we note that for the even case we can embed the set of Brauer diagrams into the set of generalized Brauer diagrams as B[d]= B[d](k,0). (5.15) 1≤ak≤d b Theorem 5.32. Assume that one of the following conditions holds: sdimV 6= 2m|0 and d ≤ m + n or (5.16) sdimV = 2m|0 with m> 0 and d